Dynamo Models for Planets Other Than Earthglatz/pub/stanley_glatzmaier_space...Dynamo Models for...

Space Sci RevDOI 10.1007/s11214-009-9573-y

Dynamo Models for Planets Other Than Earth

Sabine Stanley · Gary A. Glatzmaier

Received: 6 May 2009 / Accepted: 4 August 2009© The Author(s) 2009

Abstract Observations from planetary spacecraft missions have demonstrated a spectrumof dynamo behaviour in planets. From currently active dynamos, to remanent crustal fieldsfrom past dynamo action, to no observed magnetization, the planets and moons in our solarsystem offer magnetic clues to their interior structure and evolution. Here we review numeri-cal dynamo simulations for planets other than Earth. For the terrestrial planets and satellites,we discuss specific magnetic field oddities that dynamo models attempt to explain. For thegiant planets, we discuss both non-magnetic and magnetic convection models and their abil-ity to reproduce observations of surface zonal flows and magnetic field morphology. Futureimprovements to numerical models and new missions to collect planetary magnetic datawill continue to improve our understanding of the magnetic field generation process insideplanets.

Keywords Planetary magnetic fields · Planetary dynamos · Numerical dynamo models

PACS 91.25.Cw · 91.25.Za · 96.12.Hg · 96.12.Pc · 96.15.Gh · 96.15.Nd

1 Introduction

In the past four decades, magnetometers have flown on missions to every planet in the solarsystem. Magnetic field observations have demonstrated the diversity of planetary magneticfields and provided vital information on planetary interiors. Earth, Jupiter, Saturn, Uranus,Neptune, Ganymede and most likely Mercury have active dynamos generating magnetic

S. Stanley (�)Department of Physics, University of Toronto, 60 St. George St., Toronto, ON, M5S1A7, Canadae-mail: [email protected]

G.A. GlatzmaierEarth and Planetary Sciences Department, Earth and Marine Sciences Building,University of California, Santa Cruz, CA 95064, USAe-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

S. Stanley, G.A. Glatzmaier

fields. Earth, Moon, Mars, and possibly Mercury, have remanent crustal fields due to dy-namo action in their ancient pasts. For more details on planetary magnetic fields, we referthe reader to papers by Hulot et al., Langlais et al., Dougherty and Russell, Jia et al., andAnderson in this issue.

Over the past two decades, numerical dynamo simulations have been used to investigateplanetary magnetic fields. Although the majority of these simulations were investigations ofEarth’s dynamo, studies of other planetary dynamos have flourished in recent years. In thispaper, we review dynamo models for planets other than Earth. We discuss the differencesin planetary magnetic fields which modelers attempt to explain as well as the differencesin planetary interiors which need to be included in the models. We will concentrate on theaspects of dynamo modeling that are pertinent to our objectives here. For further details onthe theory and modeling of dynamos, we refer the reader to the papers by Wicht and Tilgnerand by Christensen in this issue.

Numerical dynamo models discretize and evolve the equations governing magnetic fieldgeneration in an electrically conducting fluid spherical shell. The system is governed by thefollowing equations:

– Conservation of mass:

Dρ

Dt= −ρ(∇ · u) (1)

where ρ is density, D/Dt is the Lagrangian derivative, and u is velocity.– The magnetic induction equation:

∂B∂t

= ∇ × (u × B) − ∇ × (λ∇ × B) (2)

where B is magnetic field, and λ = (σμ0)−1 is the magnetic diffusivity where σ is elec-

trical conductivity and μ0 is the magnetic permeability.– The momentum equation:

ρDuDt

+ 2ρ� × u = −∇P + ρg + J × B + ∇ · τ (3)

where � is the angular velocity of the rotating frame of reference, P is pressure, g is thegravitational acceleration, J is current density, and τ is the deviatoric stress tensor.

– A prescribed equation of state (EOS):

ρ = ρ(T ,P ) (4)

where T is temperature. The EOS is typically chosen to be the liquid EOS for the fluidcores of terrestrial planets and the perfect gas EOS for the atmospheres and shallow inte-riors of giant planets.

– The energy equation:

ρCp

DT

Dt− αT T

DP

Dt= J 2

σ+ τ : ∇u + H − ∇ · q (5)

where CP is the specific heat at constant pressure, αT is the coefficient of thermal expan-sion, H is the volumetric heat sources, and q is heat flux.


Aside from assuming a single component fluid, we have kept the equations quite general.All planetary dynamo models solve these equations, but different approximations, boundaryconditions, stability and parameter regimes may be used in simulating the different planets.

The timescales associated with fast acoustic and seismic waves in the core are not impor-tant for dynamo processes. Therefore, models usually filter out these waves by making eitherthe Boussinesq or anelastic approximation, both of which take ∂ρ/∂t = 0 in the conservationof mass equation and instead, solve for the evolution of density via (3)–(5). The Boussinesqapproximation goes further by assuming a constant density except for in the buoyancy termand also assuming that density is only a weak function of temperature. The Boussinesqapproximation is technically only valid for planets where the density scale height is muchlarger than the depth of the convective shell being considered. It is relatively acceptablefor the terrestrial planets. For example, the largest terrestrial planet, Earth, has ro ≈ 0.4HT

where HT = CP /αT g is the average density scale height in the outer fluid core and ro is thecore radius.

For the giant planets, the anelastic approximation is the better approximation since theconvective shell thicknesses can be much larger than the scale height. However, some gi-ant planet dynamo models use the Boussinesq approximation as a simplification if they areinterested in understanding certain mechanisms or features of the magnetic field that maynot be a direct consequence of the density stratification of the fluid. The local density scaleheight is also a strong function of radius, being orders of magnitude larger in the deep inte-rior than near the surface of a giant planet. Therefore, it can be argued that the Boussinesqapproximation is justified in the very deep interior of a giant planet.

In the following sections, we discuss various planetary dynamo models. Section 2 dealswith the two terrestrial planets other than Earth for which we have evidence of current or pastdynamo action. The dynamo models for these planets are usually concerned with explaininganomalous (i.e. non-Earth-like) magnetic observations and the solutions usually appeal tosome combination of core geometry, convective stability or mantle influence. Because wehave no observations of time variation or core fluid flows for these terrestrial planets, theonly core properties we can compare are magnetic field strength and morphology. Luckily,these two characteristics present enough puzzles to keep dynamo modelers busy. In Sect. 3we review both non-magnetic and magnetic convection models for the giant planets. Theseplanets provide us with additional model constraints in the form of observations of fluidflows in the outer layers of the planet. In Sect. 4 we discuss the known satellite dynamosand in Sect. 5 we conclude with a discussion of the current status and future of planetarydynamo modeling.

2 Terrestrial Planet Dynamos

The four terrestrial planets present a complete spectrum of dynamo states. Earth has apresently active dynamo generating an axial-dipole dominated field and paleomagnetic stud-ies demonstrate that the dynamo has been around for at least the last three billion years. Incontrast, Earth’s sister planet Venus does not presently have an active dynamo, nor any ev-idence for remanent crustal fields implying dynamo action in its past, although better datawill be required to completely rule out crustal magnetism on Venus. It is surprising that thesetwo planets, so similar in internal structure and composition, present such different dynamostates. This difference is most likely due to the planets’ abilities to cool their cores, clearlydemonstrating the importance of the mantle in driving the dynamos.


The two smaller terrestrial planets, Mercury and Mars, have evidence for dynamo actionsometime in their histories (Mercury currently and Mars in its early history). Both plan-ets present magnetic mysteries which dynamo modelers must address and we discuss eachplanet individually below.

2.1 Mercury

Mercury’s magnetic field was observed during the first and third flybys of the Mariner 10mission in the mid-1970’s (Ness et al. 1975, 1976). Recent flybys of Mercury by the MES-SENGER spacecraft have confirmed these measurements and fit the observations with a di-pole moment of ∼ 230–290 nT-R3

M , where RM is Mercury’s radius (Anderson et al. 2008).Little information on Mercury’s magnetic field morphology is presently available aside fromthe dipole moment and possibly, the quadrupole moment, but more constraints are likelyonce MESSENGER begins taking measurements from orbit in 2011. Libration observationsdemonstrate that Mercury’s core is at least partially liquid, suggesting that the magnetic fieldmay be the result of an active dynamo (Margot et al. 2007). Another possibility, that the fieldis a remanent crustal field, requires lateral inhomogeneities in the crust (Stephenson 1975;Srnka 1976; Aharonson et al. 2004). Due to the purpose of this article, we will only considerthe active dynamo possibility here.

If we assume Mercury’s dynamo works much like Earth’s dynamo, we can estimate theexpected strength of the resulting magnetic field in the core. Independent scalings basedon energetics and magnetostrophic balance suggest that Mercury’s dynamo would producea core magnetic field intensity of the order 105–107 nT (Stevenson 1987; Schubert et al.1988). Assuming an Earth-like field partitioning between toroidal and poloidal magneticfields, this implies Mercury’s surface field should be of order 4 × 103–4 × 106 nT, muchstronger than the observed value of ∼ 260 nT.

Dynamo models for Mercury have focused on explaining the weakness of the observedfield. Some reasoning or mechanism is proposed for why Mercury’s dynamo should be non-Earth-like, and models are constructed to demonstrate that the proposed mechanism canproduce a weaker Mercury-like surface field.

Several models evoke core geometry in their reasoning. Stanley et al. (2005) demon-strate that dynamos operating in thin shells with sufficiently low Rayleigh numbers suchthat convection only onsets outside the tangent cylinder can produce weak surface fields.In these models, toroidal fields are maintained strong throughout the core by differentialrotation between the inner and outer cores, but poloidal fields are inefficiently regeneratedby the weak convection in the limited area outside the tangent cylinder (Fig. 1(a)). This re-sults in a weaker observed surface field. This model allows the generation of a strong fielddynamo, but a non-Earth like field partitioning between poloidal and toroidal fields, therebyexplaining why the observed (poloidal) fields are weaker than anticipated.

Takahashi and Matsushima (2006) have also published thin-shell dynamo models thatproduce weak dipole surface fields, but the reason for the weak fields is different. In theirmodels, when a sufficiently large Rayleigh number is used such that convection is vigorousinside the tangent cylinder, smaller-scale non-dipolar fields dominate in the core. Becausethe power in smaller-scale modes decreases with distance faster than larger-scale modes, thesmall scale modes are much weaker at the surface and the observed surface field is dom-inated by its weak dipole component (Fig. 1(b)). Although this appears to be a promisingmechanism, one issue with the models is that the inner core is made electrically insulating.The authors state that when they use conducting inner cores in their models, stronger dipolarfields result, suggesting that Mercury may not be explainable with this mechanism.


Fig. 1 Mercury dynamo model geometries. (a): Stanley et al. (2005) thin shell model with convection onlyoutside the tangent cylinder, (b) Takahashi and Matsushima (2006) thin shell model with convection through-out the core, (c) Heimpel et al. (2005a) very thick shell model with isolated convection plume, (d) Christensen(2006) surrounding stable layer model, (e) Vilim et al. (2008) double dynamo model. A meridional slice isshown in (a) and (b) where the vertical line is the rotation axis, (c–e) are equatorial slices. The solid inner coreis shown in green, stably-stratified layers are shown in blue and convectively unstable layers are shown inpink. In (a) and (b), long vertical cylinders represent the convection rolls outside the tangent cylinder whereassmaller circles represent the more 3-D convection pattern inside the tangent cylinder. In (c)–(e) slices of theconvection rolls are represented as ellipses

At the other extreme in geometry, Heimpel et al. (2005a) rely on a very thick shell geom-etry to explain Mercury’s weak observed field. Their numerical dynamo simulations whichincorporate a very small solid inner core result in a single-plume mode of convection. Thisweaker convection generates weaker poloidal magnetic field explaining Mercury’s weak ob-served surface field (Fig. 1(c)).

Other models for Mercury’s weak surface field employ stable stratification (Christensen2006; Christensen and Wicht 2008). Observations of Mercury’s surface heat flow suggest itis possible that the outer portion of Mercury’s core is thermally stratified and hence stableto convection. Numerical models which employ a thick outer stable layer in the core pro-duce a weak surface field even though the field in the deep dynamo source region is strong(Fig. 1(d)). This is because the surrounding stable layer acts to attenuate the field through theskin effect, preferentially removing smaller-scale fields. Due to the slower rotation rate inMercury, scaling laws suggest that Mercury’s dynamo should be in the non-dipolar regime.Therefore, these models are affected both by the faster decay of the smaller scale features(since the dynamo operates deeper in the core) and the skin effect due to the stable layer.

Vilim et al. (2008) presents another Mercury model which relies on a non-Earth-likecore stratification. Recent experiments have demonstrated that the Fe-S system producesnon-ideal behaviour at Mercury-like pressures and temperatures for sulfur concentrations inthe range 7–12 wt% (Chen et al. 2008). This non-ideal behaviour results in the dissolutionof Fe from S at various locations in the core depending on the sulfur concentration. For


example, it is possible that Mercury’s core is being driven by the freezing out of an Fe snowat either the outer boundary of the core or mid-way through the core, or both. If the layeris at mid-depth, then the freezing out of Fe also results in the release of a more buoyant Srich fluid above the layer which can also drive convection. Dynamo models operating in thisgeometry demonstrate that it is possible to reproduce Mercury’s weak magnetic field throughthis mechanism (Fig. 1(e)). The two dynamo regions (one above and one below the mid-freezing layer) can drive dynamos that produce fields of opposite sign thereby diminishingthe observed surface field strength.

Another model proposes a magnetospheric feedback mechanism to explain Mercury’sweak surface field (Grosser et al. 2004; Glassmeier et al. 2007a, 2007b; Heyner et al. 2009).Because Mercury has a small magnetosphere (owing to its weak magnetic field) and isdeeply embedded in the solar wind (owing to its proximity to the Sun), magnetosphericcurrents can generate significant magnetic fields affecting dynamo action in Mercury’s core.Chapman-Ferraro currents, resulting from interactions between the internal dynamo and thesolar wind, can generate an ambient field of order 50 nT at Mercury’s surface. Kinematic dy-namo models show that this ambient field is opposite in polarity to the field generated in thecore, resulting in cancellation of field in the core. This mechanism is termed a “feedback”because the small magnetosphere results in these opposite polarity ambient fields whichresults in perpetuating the weak magnetosphere.

A thermoelectric dynamo was an early proposed alternative to the normal dynamo expla-nation for Mercury (Stevenson 1987; Giampieri and Balogh 2002). In this model, topogra-phy on the core-mantle boundary results in a poloidal thermoelectric current and associatedtoroidal field. Helical motions due to weak convection then act on the produced toroidalfield to generate the weak observed poloidal field. Since this model does not rely on con-vection alone to drive the dynamo, it doesn’t suffer from the same large scaling estimatesfor the intensity of the produced field. However, it does require good electrical conductivityof the lower mantle for the thermocurrents to close. It would be interesting to see numericalsimulations of a thermoelectric dynamo to determine the feasibility of explaining Mercury’smagnetic field.

When Mercury’s magnetic field was first observed, it was difficult to explain the fieldthrough a dynamo mechanism. It now appears that several mechanisms are possible, so thecurrent problem is distinguishing which mechanism is operating in Mercury. DeterminingMercury’s inner core size from upcoming observations will be challenging and so it will bedifficult to eliminate potential dynamo mechanisms by gathering independent (not related tomagnetic field) data on Mercury’s core geometry. However, it may be possible to distinguishbetween the different mechanisms by examining characteristic features of the resulting mag-netic field such as the observed surface magnetic power spectrum, locations of strong fluxspots, and secular variation. Future data from the MESSENGER (Zuber et al. 2007) andBepi-Colombo (Wicht et al. 2007) missions may therefore be able to distinguish betweenthe models.

The dynamo models make various predictions for the surface magnetic power spectrum.The Stanley et al. (2005) thin shell model and Heimpel et al. (2005a) thick shell modelproduce significant power in the dipole and quadrupole components whereas the Vilim et al.(2008) double-dynamo model produces larger octupole power compared to the quadrupolemode. The Takahashi and Matsushima (2006) model produces strong power in several non-dipolar modes, whereas the Christensen (2006) model has little power in any mode but thedipole and quadrupole. The thermoelectric dynamo power spectrum depends on the scale oftopography on the CMB.

Information may also be gathered by the location and secular variation of small-scaleflux spots. Small scale intense flux spots are limited to equatorial regions in the Stanley et


al. (2005) and Vilim et al. (2008) models. In contrast, the Heimpel et al. (2005a) thick shellmodel produces strong normal flux spots at higher latitudes. These models also experiencesecular variation. The Christensen (2006) model suggests the fields should have little energyin higher harmonics and very slow secular variation due to the skin effect of the surroundingstable layer.

2.2 Mars

Mars’ remanent crustal magnetic field, globally mapped by Mars Global Surveyor (Acuna etal. 1999), is most likely the result of an active dynamo in Mars’ early history. Several mys-teries surround the field including its hemispheric difference in field strengths, the magneticcarriers in the rocks and the driving force and timing of the dynamo. It is most likely thatthe dynamo was active sometime between 4.5–3.9 Ga since the large impact basins fromthe late-heavy bombardment (∼ 3.9 Ga) are demagnetized, the majority of Tharsis (< 3.9Ga) is demagnetized, and the ancient Martian meteorite ALH84001 has a magnetization ageof 3.9–4.1 Ga (Weiss et al. 2002). Thermal evolution models have demonstrated that Marscould have maintained a super-adiabatic temperature gradient for millions to hundreds ofmillions of years sometime before 3.9 Ga if, for example, the core was initially superheated(Breuer and Spohn 2003; Williams and Nimmo 2004), magma ocean overturn resulted inthe placement of cold cumulates at the core-mantle boundary (Elkins-Tanton et al. 2005), orplate tectonics was active in Mars’ early history (Nimmo and Stevenson 2000).

If Mars possessed a super-adiabatic temperature gradient, then a past thermal dynamois feasible. Other motions such as compositional convection as the inner core freezes andreleases a light element (Stevenson 2001) or tidal motions due to interactions of Mars withan orbiting body (Arkani-Hamed et al. 2008) do not require a super-adiabatic core, furtherbroadening the range of parameters for which an active dynamo could exist on early Mars.Martian interior models show that the core radius is approximately half the planetary radius,but aside from tidal Love numbers suggesting that some portion of the core is still liquid(Yoder et al. 2003), no information on the size of the solid inner core is available. SinceMars’ dynamo has not been active since its early history, its likely that it either has no innercore, or that the inner core is no longer growing.

Mars dynamo models are geared towards understanding the mysteries surrounding Mars’remanent magnetic field. A recent model addresses the difference in field intensity betweenthe hemispheres (Stanley et al. 2008). Maps of the crustal magnetism demonstrate that theintensity of the fields correlate with the crustal hemispheric dichotomy. This suggests thatwhatever process produced the dichotomy in crustal thickness may be related to the di-chotomy in magnetic fields. Both endogenic and exogenic formation scenarios for the crustaldichotomy have been presented in the literature. In the endogenic models, a degree-1 man-tle circulation is responsible for the different hemispheric crustal thicknesses. This degree-1pattern can result from a variety of scenarios such as mantle phase transitions (Weinstein1995; Harder 1998; Breuer et al. 1998), radial viscosity variations in the mantle (Zhong andZuber 2001; Roberts and Zhong 2006), magma ocean overturn (Elkins-Tanton et al. 2003,2005) or boundary layer instabilities resulting from a superheated core (Ke and Soloma-tov 2006). In the exogenic models, a large glancing impact in the northern hemisphere isresponsible for the excavation of northern hemisphere crust (Wilhelms and Squyres 1984;Frey and Shultz 1988; Andrews-Hanna et al. 2008; Marinova et al. 2008; Nimmo et al.2008).

Both the endogenic and exogenic dichotomy formation scenarios have implications forthe temperatures at Mars’ core-mantle boundary (CMB). In the endogenic models, if the


Fig. 2 (a): The degree-1 fixed heat flux boundary condition for the Stanley et al. (2008) Mars dynamo model.The superadiabatic heat flux is largest out of the southern hemisphere. Units are W/m2 and a positive directionis into the core (hence outward heat flux at the CMB is negative). A thermal conductivity k = 40 W/mK wasused to dimensionalize the superadiabatic heat flux in the models, however, dimensionalization is somewhatmisleading due to the numerically unavoidable parameter regime used in the simulation. (b) The surfaceradial magnetic field for the Stanley et al. (2008) Mars dynamo model. Units are in nT

upwelling manifests in the northern hemisphere (resulting in basal crustal erosion and hencea thinner northern hemisphere crust) then the temperature on Mars’ northern hemisphereCMB is hotter than in the southern hemisphere (where cold mantle material is falling ontothe CMB). In the exogenic models, thermal perturbations due to the large impacts in thenorthern hemisphere can penetrate to the CMB under the impact (Watters et al. 2009). Bothdichotomy formation scenarios can therefore result in a larger heat flux from the southernhemisphere CMB compared to the northern hemisphere CMB.

Stanley et al. (2008) implement the degree-1 CMB heat flux boundary variations impliedby the dichotomy formation scenarios in a numerical dynamo model to determine the effectson the magnetic field (Fig. 2(a)). The result is a single-hemisphere dynamo where magneticfield generation is concentrated and strongest in the southern hemisphere. Due to the reducedheat flux from the northern CMB, the northern hemisphere of the core is sub-adiabatic andconvection is most strongly driven in the southern hemisphere. The strong thermal winds inthe model break the Proudman-Taylor constraint (Proudman 1916; Taylor 1917), although inthe southern hemisphere, signs of rotational influences are still present. The resulting surfacemagnetic field is non-dipolar, non-axisymmetric and stronger in the southern hemisphere(Fig. 2(b)). This suggests that the reason Mars’ crustal magnetic field is concentrated in thesouthern hemisphere may be because the dynamo produced strongest fields in the southernhemisphere.

The cessation of the Martian dynamo is another mystery that dynamo modelers haveaddressed. It is generally assumed that dynamo action ceased when the driving forces weak-ened enough such that the Rayleigh number became sub-critical. However, dynamo analyt-ics (Childress and Soward 1972) show that a sub-critical dynamo is possible in the “strongfield” regime where the Coriolis, Lorentz and buoyancy forces balance in the momentumequation. Essentially, the inhibiting effects of rotation and magnetic fields can offset eachother resulting in convection at Rayleigh numbers below the critical value if only rotationwere present. Recently, Kuang et al. (2008) produced sub-critical dynamo action in a spher-ical shell model suggesting that Mars’ dynamo may have lasted longer than age estimatesbased on requiring a super-critical Rayleigh number. Their models also demonstrate thatonce the driving force decreases below the threshold for dynamo action in the sub-criticalregime, it is unlikely to start up again unless the Rayleigh number increases by about 25 per-cent, suggesting it is hard to restart the dynamo. Their Mars dynamo, which maintains arelatively strong axial-dipolar dominated field in the super-critical regime, becomes morenon-dipolar with frequent reversals in the sub-critical regime.


Another possibility for the termination of the Martian dynamo was suggested by Glatz-maier et al. (1999). The Mars dynamo may have ended by going through a series of dipolereversals and not completely recovering after each one. This occurs in one of their dynamosimulations that has a heat flux imposed at the CMB that is greatest at mid-latitude and leastat the equator and poles (note that this is a different CMB heat flux variability than that ofthe Stanley et al. (2008) work discussed above). After several hundred thousand years witha strong magnetic field (starting with a homogeneous CMB heat flux condition) this caseexperiences a spontaneous dipole reversal with the usual drop in magnetic dipole intensitybut fails to fully recover to the original field intensity after the reversal. Roughly a hundredthousand years later it reverses again and again recovers only to about 10% of its pre-reversalintegrated magnetic energy. The unfavorable CMB heat flux condition, which forces moreheat to be convected to mid-latitude instead of the preferred equatorial and polar regions,apparently destroys the more efficient fluid flow patterns for the dynamo mechanism.

3 Giant Planet Dynamos

All four of our solar system giant planets possess active dynamos, but the fields produced bythe gas giants are markedly different from the ice giant fields. Whereas Jupiter and Saturnpossess axial-dipole-dominated fields, Uranus’ and Neptune’s fields are dominated by mul-tipolar components without a preference for axisymmetry (Connerney 1993). Aside fromthis difference in field morphologies, other mysteries surrounding the giant planet magneticfields include:

– What mechanism is responsible for the zonal bands in the giant planet atmospheres, dothey represent deeper zonal flows that play a role in dynamo generation, and why are theydifferent for the different planets?

– Why is Saturn’s dynamo producing such an axisymmetric field?– How can Uranus and Neptune be generating magnetic fields with such low surface heat

flows?

3.1 Overview of the Different Types of Models for Giant Planets

Most studies of the internal dynamics of giant planets have focused on explaining the main-tenance of the observed surface flows and predicting the structure and extent of these flowsbelow the surface. The flows in the deep interior, where the fluid is electrically conducting,maintain magnetic fields by shearing and twisting the existing field at a rate sufficient tooffset magnetic diffusion. Since density increases with depth, convective velocity decreaseswith depth and electrical conductivity increases with depth (for our solar system giants).Therefore, magnetic induction by the flows and the resulting Lorentz forces on the fluidare mainly important within a range of depths in a giant planet where fluid velocity andelectrical conductivity are both sufficiently large.

To be credible, a dynamo simulation of convection and magnetic field generation in theouter part of a giant planet should at least produce fluid flows near the non-conducting sur-face similar to those observed on the particular giant planet being simulated. This provides avaluable constraint that, for example, those who simulate convective dynamos in terrestrialplanets do not have. To test how realistic are the flows and fields well below the surface ina dynamo simulation one can check if the total rate of entropy production by ohmic heatingwithin the convection zone ∫

(J 2/σT )dV (6)


is, on average, no larger than the rate entropy flows out at the top minus the rate it flowsin at the bottom of the dynamo region, i.e., no larger than the observed luminosity times(T −1

top − T −1bot ) (Backus 1975; Hewitt et al. 1975). Here, J is the simulated electric current

density and σ and T are the prescribed electrical conductivity and temperature, respectively.We will therefore first review some non-magnetic modeling studies of fluid flows in gi-

ant planets before discussing simulations of dynamically-consistent convective dynamos.There have been several two dimensional (2D) modeling studies of vortices in the shallowatmosphere of giant planets. However, since we are interested in planetary dynamos, herewe consider only three-dimensional (3D) global simulations. We will also focus on zonalwinds (i.e., the axisymmetric part of the longitudinal velocity) because they dominate theflow at the surface and because their extension into the semi-conducting region below thesurface plays a critical role in maintaining the toroidal magnetic field.

The surface zonal winds on our giant gas planets, Jupiter and Saturn, advect thelatitudinally-banded clouds observed on the surfaces of these planets (Sanchez-Lavega etal. 2002; Porco et al. 2003). These longitudinally-averaged winds alternate in latitude be-tween eastward directed (i.e., prograde) and westward directed (i.e., retrograde) relative toa rotating frame of reference (Fig. 3), which is chosen to be that of the global magnetic fieldbased on observed radio emissions. Note, that this reference frame does not represent themean angular velocity at the cloud tops nor the rotating frame in which the total angularmomentum of the planet vanishes. It also does not necessarily represent the average angu-lar velocity of the deep interior. It approximates the mean angular velocity of the region inwhich the dynamo operates and could have a phase velocity relative to this region. In addi-

Fig. 3 Plots of the observed zonal winds vs. latitude on Jupiter, Saturn and Uranus. Positive velocities areprograde; negative are retrograde. Hubble Space Telescope measurements (dots) of Saturn’s zonal winds from1996–2002 show that the peak equatorial jet has decreased by roughly a factor of two since the measurementsby the Voyager Mission (solid line) from 1980–1981. (Sanchez-Lavega et al. 2002; Porco et al. 2003; Hammelet al. 2005)


tion, like the magnetic fields in the sun and the Earth, it is not necessarily constant in time(Giampieri et al. 2006) because the field is maintained by time dependent magnetohydrody-namics. The equatorial jets on Jupiter and Saturn are prograde relative to this chosen frameof reference. Although the zonal wind pattern on Jupiter has been fairly constant for the pastthree decades, the intensity of the equatorial jet on Saturn has decreased by roughly a factorof two since the time of the Voyager Mission (Fig. 3). This may have been an actual changein wind speed or a change in the level of the clouds, which are measured to calculate thespeed.

The ice giants, Uranus and Neptune, also display a surface differential rotation rate inlatitude. However, unlike the gas giants and more like the zonal winds in the Earth’s at-mosphere, they have a broad retrograde equatorial jet and only one prograde zonal flow athigh latitude in each hemisphere (e.g., Hammel et al. 2005).

Many attempts have been made to explain the maintenance of these surface zonal windpatterns and predict their amplitudes and patterns below the surface, i.e., the differential ro-tation. They can be maintained by Coriolis forces resulting from a thermally-driven merid-ional circulation (i.e., a “thermal wind”) or by the transport of longitudinal momentum byradial and latitudinal flows (i.e., the convergence of Reynolds stress). The axisymmetricparts of viscous and magnetic forces also play a role; however, these forces usually inhibitdifferential rotation in planets.

Most studies of zonal winds in giant planets have come from two scientific modelinggroups: the atmospheric climate modeling community and the geodynamo and solar dynamomodeling communities. Both groups have attempted to use modified versions of their tra-ditional computer models. Modelers from the atmospheric community usually prescribe anatmosphere stratified in density and temperature based on estimates of what these are for therelevant planet. However, their global models are based on the assumption that the observedsurface winds are maintained exclusively by the dynamics within the very shallow weatherlayer. The “shallow-water” model simplifications are based on large-scale horizontal flowsand ignore the small-scale buoyantly-driven flows in radius. For reviews of (non-magnetic)climate models see, for example, Dowling (1995) and Showman et al. (2007).

By placing an impermeable lower surface just below the clouds, as is the case for theEarth, these atmospheric models completely ignore the convection in the deep interiors ofthese fluid bodies. The Galileo probe, however, measured strong zonal wind down to a pres-sure of 20 bars, well below the weather layer driven by solar insolation, with no indicationof the wind becoming weaker with depth (Atkinson et al. 1998). The existence of a globalmagnetic field provides indirect evidence of significant flows at much greater depths. Thatis, since the magnetic dipole decay time for a body with the size, electrical conductivity andtemperature of Jupiter’s deep interior is roughly only a million years, a dynamo likely existsin its electrically-conducting interior, driven by deep convection and zonal winds.

Modelers from the geodynamo community use 3D deep convection models to investi-gate the structure and maintenance of zonal winds (i.e., differential rotation) without makingshallow-water type approximations. However, most have assumed no stratification of den-sity, a fairly good approximation for the Earth’s fluid core but not for the outer regions ofgiant planets. A few studies, however, do account for both deep convection zone and largedensity stratification, similar to modeling studies that have been conducted in the solar dy-namo community for the past quarter century. Although these models capture more of thecorrect physics of the problem, they too need improvements. See, for example, Dormy etal. (2000), Kono and Roberts (2002) and Glatzmaier (2002) for reviews of the geodynamomodels.


The shallow-atmosphere community and the deep-convection community have mostlyignored each other and have seldom referenced each others papers. However, since deep-convection dynamo models are being judged on their ability to maintain banded zonal windsat their outer boundaries like those observed in the shallow cloud layers of our gas giants,it is important to discover and understand the mechanism that maintains these winds. Inparticular, it is important to know if this mechanism is seated mainly within the shallowatmosphere, as the atmospheric community claims, or mainly within the deep convectionzone as claimed by that community. A critical issue for studies of our solar system giantplanet dynamos is to know the depth to which the strong winds penetrate. Also, althoughthe outer atmospheres of our gas giants are relatively cold and have very low electricalconductivity, the outer atmospheres of the extrasolar gas giants in close orbit around theirparent stars are probably partially ionized by the stellar radiation. Therefore, the structuresof the flows in these electrically conducting atmospheres is very important for studies of thesurface dynamos of these planets.

The very different approaches and approximations to modeling the internal dynamics ofgiant planets have several subtle, but seldom discussed, effects on the respective results. Wediscuss the resulting deficiencies below as we describe some of the non-magnetic modelingresults. Then we discuss some simulations of convective magnetohydrodynamic (MHD) dy-namos for gas giants and ice giants. These dynamically self-consistent solutions of thermalconvection and magnetic field generation have only been produced using deep convectionmodels.

3.2 Atmospheric Models

Some atmospheric models of giant planets rely on a thermal wind scenario (e.g., Allison2000) for which a meridional circulation (the axisymmetric north-south flow) drives thezonal winds (the axisymmetric east-west flow) via Coriolis forces. That is, axisymmetricflow toward the equator moves fluid further away from the planet’s rotation axis, increasingits moment of inertia, and, to the extent that angular momentum is conserved, causes itsangular velocity to decrease. Likewise, flow toward a pole increases the fluid’s angular ve-locity. This naturally results in a retrograde zonal wind in the equatorial region relative to theglobal mean angular velocity (e.g., Williams 1978; Cho and Polvani 1996), as is the case forthe Earth’s lower atmosphere and for our ice giants. Note, that a latitudinally-banded patternof zonal winds with a prograde equatorial jet can be obtained by tuning a prescribed heatingfunction distributed in latitude and radius that continually nudges the temperature toward aprofile that drives the desired zonal wind pattern (e.g., Williams 2003). However, such math-ematical fits to surface observations do not provide a dynamically-consistent prediction orexplanation for the maintenance of an observed zonal wind structure.

Some of the most sophisticated gas giant modeling attempts within the atmospheric com-munity have been made using general circulation models (GCMs). These 3D global models,which were originally designed and used for studies of the Earth’s climate and weather, rep-resent a shallow spherical shell of density-stratified gas with a depth less than a percent ofthe planetary radius. They typically parameterize the radiation transfer using a Newtonianheating/cooling method, which simply nudges the local temperature toward a prescribedtemperature profile based on an assumed timescale. GCMs also have “shallow-water” sim-plifications based on the assumption that the length and velocity scales in radius are smallcompared to those in the horizontal direction. For example, they assume local hydrosta-tic equilibrium instead of solving the radial component of the full momentum equation.That is, the radial pressure gradient is forced to always exactly balance the weight of the


fluid above. Consequently, the Coriolis, buoyancy and viscous forces and the divergenceof Reynolds stress are all neglected in the radial direction. Therefore, instead of evolvingthe radial component of the fluid velocity via the momentum equation, as is done for thehorizontal components, it is calculated by determining what it needs to be to satisfy massconservation, hydrostatic balance and the equation of state once the horizontal divergenceof the horizontal flows is updated (at each grid point and time step).

This has worked sufficiently well for GCMs used to study the large scale flows in theEarth’s atmosphere. However, the Earth’s atmosphere does have an impermeable lowerboundary; the atmosphere of a giant planet, on the other hand, extends smoothly into adense, convecting interior (e.g., Bodenheimer et al. 2003; Guillot 1999). The radial velocityassociated with such a grid cell in a GCM represents the spatial average of many, muchstronger, small-scale up and down drafts within the entire cell; it is not meant to be a repre-sentative value of the actual dynamical velocity in the Earth’s atmosphere. When averagedthis way, these radial velocities are small relative to the horizontal winds. Therefore, mostGCMs also neglect the contribution to the Coriolis and Reynolds stress terms in the longi-tudinal component of the momentum equation that are due to radial velocity.

Solving for the radial flow via the divergence/convergence of horizontal flows, instead ofdriving it with buoyancy and Coriolis forces, misses critical aspects of the dynamics of deeprotating convection. This is apparent in profiles of the flow and temperature produced bygiant planet GCMs for which vortices parallel to the planetary rotation axis are not promi-nent; such vortices are a dominant feature of flows seen in laboratory experiments and 3Dsimulations of deep rotating convection. In addition, inertial oscillations, which are drivenby Coriolis restoring forces, are also poorly represented in GCMs because of the neglectof Coriolis forces in radius due to longitudinal flows and in longitude due to radial flows.The neglect of buoyancy to drive radial velocity and the neglect of Coriolis and Reynoldsstress terms involving the radial velocity have prevented GCM studies from self-consistentlyproducing prograde equatorial zonal winds.

Lian and Showman (2009) have recently produced much more promising GCM sim-ulations that model the effects of moist convection. Although moist convection is highlyparameterized in their model, it increases the vigor of the radial flows by releasing latentheat where water vapor condenses. That is, when fluid containing water vapor is advectedupward it condenses at some lower-temperature level and releases latent heat. This causesthermal expansion and horizontal divergence, which pulls up more fluid. Unlike previousGCM simulations, their model can produce, without prescribing ad hoc heating patterns,banded zonal surface flows with a prograde equatorial jet (Fig. 4) qualitatively similar tothose of our gas giants. These profiles are maintained due to the convergence of Reynoldsstress, including that due to the product of radial and longitudinal velocities, which previ-ous GCM studies have neglected. They argue that the latitudinal widths of the zonal windbands are determined by a “Rhines effect” (Rhines 1975), which predicts a local lengthscale proportional to the square root of the zonal wind speed. This arises when the advec-tion in latitude of radial vorticity is balanced by the generation of radial vorticity due to thelatitudinal gradient of the radial component of the planetary rotation rate.

Lian and Showman (2009) can also produce a retrograde equatorial jet (Fig. 4), like thoseof our ice giants, when they prescribe a larger source of water vapor, which effectively in-creases the vigor of the convective flows. The change from prograde to retrograde zonal flowin the equatorial region when convective driving increases relative to Coriolis effects hasbeen demonstrated in deep convection models and shown to also depend on the ratio of vis-cous to thermal diffusion and on the degree of density stratification (Glatzmaier and Gilman1982). Although the Lian-Showman model still suffers from the hydrostatic shallow-water


Fig. 4 Snapshots of the longitudinal velocity in GCM simulations with moist convection. Reds representprograde winds; blues retrograde. These simulations vary, from “Jupiter” to “Uranus/Neptune” by increasingthe amount of prescribed water vapor in the model. (Lian and Showman 2009)

simplifications mentioned above, it is beginning to capture some of the important dynamicsseen in 3D deep convection models.

3.3 Non-magnetic Deep Convection Models

It may be the case that the zonal winds on giant planets are confined to the shallow surfacelayers and driven by heat sources and moist convection there without significant influencefrom the deep convection below. However, hydrostatic shallow-water type models, includingGCMs, assume this from the start instead of allowing the full dynamics to demonstrateit. Deep convection models can also produce banded zonal winds with either prograde orretrograde equatorial jets, without neglecting the dynamics of the deep interior. They aredesigned to capture the 3D dynamics within a deep rotating fluid shell but not necessarilythe 2D dynamics in a shallow atmosphere. Therefore, they neglect moist convection andapproximate radiative transfer as a thermal diffusive process.

The deep convection models of the geodynamo and solar dynamo communities are whatthe atmospheric community would call “non-hydrostatic” models. That distinction is notmade in the dynamo communities because all 3D convection and dynamo models are non-hydrostatic. There was never a reason to make shallow-water type approximations for thesemodels because the convection zone depths for the geodynamo and solar dynamo are com-parable to the radii of the respective bodies, as they are for giant planets. Therefore, all three


components of the momentum equation are solved, putting the radial component of velocityon the same footing as the horizontal components. This is really needed to capture the fulldynamics of deep rotating convection; i.e., the entire Coriolis force and Reynolds stress areused and the buoyancy force drives convection.

The full representation of these forces in deep convection models captures the gen-eration of vorticity parallel to the planetary rotation axis as fluid moves relative to thisaxis. When Coriolis and pressure gradient forces nearly balance everywhere and allother forces are relatively small, the classic Proudman-Taylor theorem (Proudman 1916;Taylor 1917) predicts that rotating, incompressible, laminar fluid tends to flow within planesparallel to the equatorial plane. Within such a convective column or vortex, rising fluid gen-erates negative vorticity relative to the rotating frame of reference and sinking fluid generatespositive vorticity. This results in a prograde-propagating Rossby-like wave because positivevorticity is generated on the prograde sides of positive vortices and likewise negative vor-ticity is generated on the prograde sides of negative vortices. Since the effect is greatestnear the surface, rising fluid parcels within the vortices curve to the east and sinking curveto the west. This causes rising fluid to be correlated with eastward flow and sinking fluidto be correlated with westward flow. The resulting convergence of this nonlinear Reynoldsstress maintains a zonal wind (i.e., differential rotation) with a prograde equatorial jet nearthe surface. In such a case, a meridional circulation, much weaker than the zonal wind,is maintained by the Coriolis forces resulting from the zonal wind. If, on the other hand,buoyancy forces dominate over Coriolis forces, the curving of rising and sinking fluid ismuch less correlated and the thermal wind, with a retrograde equatorial jet, is maintained byCoriolis forces resulting from the meridional circulation (e.g., Glatzmaier and Gilman 1982;Aurnou et al. 2007). Of course this assumes convective velocities are large enough to makea relatively significant Reynolds stress. If they are not, as is the case for the geodynamo, thezonal flow in the equatorial region tends to be retrograde even if Coriolis forces dominateover buoyancy forces (e.g., Kono and Roberts 2002).

Two basic types of deep convection models have been used to study giant planets:those that ignore the background density stratification and assume a liquid equation of state(Boussinesq models) and those that account for density stratification and use a gas equationof state (anelastic models). There is a fundamental difference between the way these twotypes of models maintain differential rotation. Models that assume a constant backgrounddensity rely on the classic vortex-stretching mechanism (Busse 1983, 2002) to generate vor-ticity and maintain differential rotation. This relies on the existence of convective columnsaligned parallel to the planetary rotation axis and spanning from the outer (impermeable)surface in the northern hemisphere to that in the southern hemisphere. Negative vorticity(i.e., an anti-cyclone) is generated in rising fluid (within such a column when outside thetangent cylinder to the inner boundary) because conservation of mass and incompressibilityrequire rising fluid to spread out normal to the axis as it approaches the sloping imperme-able outer boundary. The resulting Coriolis torque generates the vorticity, which drives aprograde propagating Rossby-like wave (as described above) because of the spherical shapeof the outer boundary. Likewise, sinking fluid is stretched parallel to the axis because ofthe sloping boundaries, generating positive vorticity (i.e., a cyclone). Under the specialgeostrophic conditions mentioned above, with enough viscosity to maintain relatively lam-inar flow and large diameter columns, this mechanism is able to maintain differential rota-tion via the convergence of Reynolds stress as explained above (e.g. in simulations: Gilmanand Miller 1981; Christensen 2002; Heimpel et al. 2005b; and in laboratory experiments:Busse and Carrigan 1976; Hart et al. 1986; Manneville and Olson 1996; Aubert et al. 2001;Aurnou 2007).


Fig. 5 A snapshot of thelongitudinal winds maintained onthe outer and inner boundaries ina simulation of Jupiter using aBoussinesq, deep rotatingconvection model. Reds representprograde winds; blues retrograde.(Heimpel et al. 2005b)

Most published studies of convection in giant planets have been based on the Boussinesqapproximation and most have simulated only the outer 10 to 20% in radius of the planet(see papers by Wicht and Tilgner and by Christensen in this issue). The resulting differen-tial rotation manifests itself on the surface as a zonal wind, beautifully-banded in latitude,with a strong prograde equatorial jet qualitatively similar to those of Jupiter and Saturn.An example by Heimpel et al. (2005b) is shown in Fig. 5.

Heimpel et al. (2005b) argue that the widths of the bands in latitude are determined bya modified “Rhines effect” due to the approximate balance of the advection of vorticity byzonal wind and the stretching of fluid along the axes of convective columns due to the spher-ical geometry of the outer impermeable boundary. This Coriolis stretching effect decreaseswith latitude, unlike the Coriolis effect of shallow-water GCMs. Therefore, this mechanismmaintains a prograde equatorial jet. According to the authors, the location of the inner im-permeable boundary in their model determines the width and amplitude of the equatorialjet.

In addition to the prograde equatorial jets of Jupiter and Saturn, the retrograde equato-rial jets of Uranus and Neptune have been investigated using Boussinesq deep convectionmodels. Aurnou et al. (2007) show that the direction of the equatorial jet in their models de-pends on the relative importance of Coriolis and buoyancy forces as Glatzmaier and Gilman(1982) demonstrated with anelastic models. As discussed above, when Coriolis forces dom-inate over buoyancy forces a prograde equatorial jet, similar to those seen on the gas giants,is maintained. If instead buoyancy forces become comparable to or exceed Coriolis forces,a retrograde equatorial jet is maintained.

Although beautiful results have been obtained by these deep-convection constant-densitysimulations, an obvious question is how realistic is the simulated structure of the convectionand the maintenance of differential rotation when density stratification is ignored. The den-sity in giant planets varies in radius by many orders of magnitude, especially in the outerpart of the planet (e.g., Bodenheimer et al. 2003; Guillot 2005). For the Boussinesq ap-proximation to be valid, the depth of the convection zone needs to be small relative to the


local density scale height. However, since the top of the convection zone is at a pressureof roughly 1 bar, Jupiter’s outer 10% in radius, for example, spans about eight density scaleheights. That is, the expansion of rising fluid and contraction of sinking fluid have first-ordereffects on the fluid flows in the outer regions of giant planets. In addition, because the interi-ors of giant planets have very small viscosity, convective columns (vortices) in these planetslikely have extremely small diameters and the flows are probably strongly turbulent, notlaminar. Under these conditions, it is unlikely that such long thin convective columns woulddevelop and stretch uniformly while spanning many density scale heights from the northernto southern boundaries (Glatzmaier et al. 2009). Instead, many short, disconnected vorticesor vortex sheets would likely exist, on which the sloping outer boundary would have littleeffect.

To capture the effects of a significant density stratification and a gas equation of state, forwhich density perturbations depend on both temperature and pressure perturbations, somedeep convection models employ the anelastic approximation. As for GCMs and Boussinesqmodels, sound waves are naturally filtered out in anelastic models in order to use much largernumerical time steps. This approximation to the fully compressible system of equations isvalid when the fluid velocity is small compared to the local sound speed and thermody-namic perturbations are small relative to the mean (spherically symmetric) profiles, as islikely the case within giant planets. Background (reference state) profiles in radius of den-sity, temperature and pressure are prescribed that are hydrostatic and usually adiabatic; thisreference state may also evolve in time. The 3D time-dependent thermodynamic perturba-tions, which are neither hydrostatic nor adiabatic, are solved relative to this state. Variousinitial reference states have been employed: for example, polytropic solutions to the Lane-Emden equation (e.g., Hansen and Kawaler 1994) and polynominal fits to one-dimensionalevolutionary models of giant planets (e.g., Guillot 2005). The equation of state for the outerpart of a gas giant can be approximated as a perfect gas, i.e., reference state pressure isproportional to density times temperature. However, the deep interior is a non-relativisticdegenerate electron gas (e.g., Guillot et al. 2004), with reference state pressure proportionalto density to the 5/3 power (e.g., Hansen and Kawaler 1994).

Recently Jones and Kuzanyan (2009) produced anelastic simulations to study the inter-nal convection and differential rotation of density-stratified gas giants. In one of their studiesthey compare solutions in the outer 30% in radius; one case spans five density scale heights(i.e., a factor of 148 in density) and another spans only a tenth of a scale height (essentiallyBoussinesq). Both of these cases have a broad equatorial prograde jet, with relatively lit-tle zonal wind at higher latitudes (Fig. 6). They clearly demonstrate the effects of densitystratification. In the strongly stratified case the amplitude of the convection and differentialrotation decrease significantly with depth; whereas the amplitudes are much more uniformin depth for the nearly constant density case.

One of us (Glatzmaier) has simulated the internal dynamics of giant planets using amodified version of an anelastic solar dynamo model (Glatzmaier 1984). Figure 7 showssnapshots of two recent anelastic simulations that use the same density background profile,which spans five density scale heights. These two cases also have the same thermal diffusiv-ities and the same velocity and thermal boundary conditions. However, viscous diffusivityfor case 2 is ten times smaller than that for case 1. This decrease in viscosity increases bothbuoyancy and Coriolis forces by roughly the same factor. The more viscous case (1) main-tains a shallow prograde jet in the equatorial region with a peak velocity of 140 m/s (relativeto the rotating frame) and essentially solid-body rotation inside a cylinder tangent to this jet(i.e., not tangent to the inner impermeable boundary); the amplitude of the convection dropstwo orders of magnitude from the outer to the inner regions. The more turbulent case (2) has


Fig. 6 Snapshots of the radial component of the convective velocity in the equatorial plane (left) and thezonal winds in the meridian plane (right) for two anelastic simulations. The top row is for a case that isnearly Boussinesq (spanning 0.1 density scale heights). The bottom row is a case with a significant densitystratification (spanning 5 density scale heights). On the right, green is upflow and blue is downflow. On theleft, red is prograde and blue is retrograde. (Jones and Kuzanyan 2009)

a deeper equatorial jet, which peaks at 470 m/s at the surface, and several weaker alternatingzonal wind bands that penetrate the deep interior and extend to the poles; the convectiveamplitude drops only about one order of magnitude with depth. Jones and Kuzanyan (2009)see qualitatively similar results. Starchenko and Jones (2002) argue that the amplitude ofthe convective flows deep within gas giants should be about a cm/s or less in order to main-tain roughly the same total convective heat flow as observed at the surface. One of theirassumptions is that rising fluid is always hot and sinking is cold, which is reasonable forlaminar convection. However, for strongly turbulent convection, especially when dominatedby vortices, this need not be the case. Therefore, one could expect the average amplitude ofthe flows to increase, without increasing the net heat flow, when viscosity is decreased. Thiseffect, however, may be greater near the surface where the flow is more turbulent.

Although long, thin and straight convective columns spanning the entire interior arenot precluded in these anelastic simulations, they seldom develop when the flow is evenweakly turbulent. Unlike the Boussinesq models, however, such structures are not needed


Fig. 7 Snapshots of the radial component of the convective velocity in the equatorial plane (left) and thezonal winds in the meridian plane (right) for two anelastic simulations, both spanning five density scaleheights. Case 2 has viscosity ten times smaller than that of Case 1. Reds and yellows are upflow (left) andprograde (right); blues are downflows (left) and retrograde (right). (Simulations by G.A. Glatzmaier)

to maintain a differential rotation pattern with a prograde equatorial jet at the surface.That is, instead of being generated by a stretching torque, local vorticity (parallel tothe planetary rotation axis) is generated mainly by compressional torque due to the ex-pansion of rising fluid and contraction of sinking fluid (Glatzmaier and Gilman 1981;Glatzmaier et al. 2009). A prograde propagating Rossby-like wave is now driven becausedensity decreases outward, not because the boundaries are spherical; and the persistent tilt inlongitude of rising and sinking fluid, required for the convergence of prograde angular mo-mentum in the surface equatorial region occurs because the density scale height decreaseswith increasing radius (at least in the gas giants). This mechanism for maintaining differen-tial rotation exists even for short isolated vortices rising and sinking within strongly turbulentenvironments. The resulting pattern of differential rotation, in radius and latitude, dependson the details of the thermal, compositional and density stratifications in radius.

One important distinction to recognize is the difference between these non-axisymmetricconvective vortices and the axisymmetric zonal flows centered on the planetary rotation axis(i.e., the “constant-on-cylinders” differential rotation). The former tend to be relatively small


scale; whereas the latter are broad and, at least in the equatorial region, do extend throughthe interior from the northern surface to the southern surface. This difference exists becausethe zonal flow has no component in radius and therefore does not experience the densitystratification.

3.4 Magnetic Deep Convection Models

3.4.1 Gas Giants

Having discussed the various 3D simulations of convective structures and zonal flows thatmay occur in giant planets, we now consider the types of magnetic fields these flows ofelectrically-conducting fluid could maintain by shearing and twisting existing fields. Mod-els of the geodynamo and solar dynamo typically prescribe a constant electrical conductiv-ity. Several computational studies of dynamos in gas giants have also assumed a constantbackground density and electrical conductivity (see papers by Wicht and Tilgner and byChristensen in this issue). However, more realistic models of gas giants need to account forthe large stratifications in density and electrical conductivity in the outer semi-conductingregion, which have first order effects on the structure of the generated magnetic field. Theelectrical conductivity, for example, likely increases exponentially with depth before be-coming constant below the molecular-metallic hydrogen phase transition (e.g., Guillot et al.2004). That is, in the outer semi-conducting region magnetic diffusivity is not constant, butincreases rapidly with radius (Liu et al. 2008).

The effects of significant stratifications in electric conductivity and density have beendemonstrated in an anelastic dynamo simulation of a Jupiter-like planet (Glatzmaier 2005).This deep convection zone model spans the outer 80% in radius; the inner 20% is assumedto be a solid core. Constant viscous and thermal diffusivities are prescribed, with the ratioof viscous to thermal being 0.01. The prescribed electrical conductivity increases exponen-tially with depth by three orders of magnitude in the outer semi-conducting region, up to aconstant value in the lower “metallic hydrogen” region. However, the amplitude of convec-tion decreases with depth because of the density stratification. Therefore, magnetic energy isvery small in the deep interior and instead peaks in the lower semi-conducting region, whereelectrical conductivity and convective velocity are both high enough to efficiently generatemagnetic field. The results are sensitive to the number of density scale heights represented,the radial profile of electrical conductivity, the vigor of the turbulent convection and therelative effects of Coriolis forces.

Guided by this deep-shell anelastic simulation, Glatzmaier (2005) also simulated a shal-lower convective dynamo to study the dynamics of the semi-conducting region. Only theouter 20% of the planetary radius is simulated. This generic gas-giant model has a radius, arotation rate and radial profiles of density, temperature and pressure that are roughly aver-ages of what are obtained in one-dimensional models of Jupiter and Saturn (Guillot 2005).In terms of traditional non-dimensional parameters, the (classic) Rayleigh number is 109,Ekman number is 5 × 10−7 and Prandtl number is 3 × 10−2. The simulation spans about6 years, more than 3,000,000 numerical time steps, the last half of these at a spatial resolu-tion of 768 grid points in latitude, 768 in longitude and 241 in radius.

Due to the density stratification, the convective velocity near the transition depth is typi-cally an order of magnitude smaller than it is near the surface and two orders of magnitudesmaller than the surface zonal winds. Small-scale isolated vortex structures exist in the con-vection zone. Differential rotation persists throughout the convection zone as latitudinally-alternating angular velocity constant on cylinders coaxial with the rotation axis (Fig. 8).


Fig. 8 Snapshots of an anelastic simulation of a gas giant dynamo. Top row: The longitudinal component ofvelocity at the surface, its zonal average vs. latitude and the zonal velocity displayed in the meridian plane.Reds and yellows are eastward up to 300 m/s; blues are westward relative to the rotating frame up to 100 m/s.Bottom row: The radial component of the magnetic field at the surface, its zonal average vs. latitude and the3D field illustrated with lines of force. Reds and yellows are outward directed; blues are inward. Orange linesof force are directed outward; blue are directed inward. Typical intensity within the convection zone is a fewhundred gauss. (Glatzmaier 2005)

This appears on the surface as a banded axisymmetric zonal wind profile. The differentialrotation in radius is maintained by the density-stratification mechanism (Glatzmaier et al.2009) because the density scale height decreases with radius, as discussed above. The peakzonal wind velocities at the surface decrease with latitude because the density scale height,measured normal to the rotation axis, naturally decreases with latitude. The strength andlatitudinal extent of the equatorial jet at the surface (Fig. 8), which here is more like Sat-urn’s than Jupiter’s (Fig. 3), is sensitive to the prescribed density stratification, viscosityand the ratio of buoyancy to Coriolis forces. Maintaining a differential rotation profile morelike Jupiter’s will likely require a more turbulent simulation, which would be achieved bydecreasing viscosity while increasing spatial and temporal resolution.

Although convection kinetic energy peaks near the surface in this simulation, magneticinduction is most efficient in the lower quarter of the simulated convection zone where theelectrical conductivity is greatest. There the differential rotation shears the poloidal field intotoroidal (east-west) fields. To limit this shear (and the resulting ohmic dissipation), poloidalfield tends to align parallel to surfaces of constant angular velocity, as suggested by Ferraro’siso-rotation law (Ferraro 1937). This makes the radial field at the surface significantly larger


Table 1 Surface magnetic field structures. A selection of the low-degree (�) axisymmetric (m = 0) Gausscoefficients (in units of gauss) for the magnetic fields of Jupiter (Connerney et al. 1998), Saturn (Giampieriand Dougherty 2004) and the simulation by Glatzmaier (2005) (averaged in time). The sign of the axial dipole(g0

1) defines the dipole polarity

Gauss coefficients

� Jupiter Saturn Simulation

g0�

g0�

g0�

1 4.205 0.212 −2.770

2 −0.051 0.016 −1.041

3 −0.016 0.028 0.715

4 −0.168 −1.058

at high latitudes, as seen in Fig. 8. Also seen is the banded pattern of the radial componentof the field at the surface, related to the banded pattern of the angular velocity.

The simulation’s dipole moment is 1.1 × 1020 Tm3, somewhat less than Jupiter’s(1.5 × 1020 Tm3) and larger than Saturn’s (4.2 × 1018 Tm3). More detailed structure ofthe magnetic field at and beyond the surface is usually described in terms of Gauss coeffi-cients in a spherical harmonic expansion (e.g., Connerney et al. 1998). Table 1 lists a few ofthe lowest degree, �, coefficients (i.e., largest scales) from fits to observations of the fieldsof Jupiter and Saturn and from the snapshot of the simulated field illustrated in Fig. 8. Betterobservations are needed to describe the planetary fields beyond degree 4 (see the paper byDougherty and Russell in this issue); the computer simulation, in comparison, updates over130,000 Gauss coefficients every numerical time step. Only the axisymmetric coefficients(m = 0) are listed here since the non-axisymmetric coefficients depend on the choice ofthe longitude zero and are more time dependent. However, it is seen that even the first fouraxisymmetric coefficients for the simulation do not agree with those for either planet, otherthan being of the same order of magnitude and having an axial dipole (g0

1) that dominates.Note that its sign merely reflects the dipole polarity, which for the simulation is currentlyopposite of the present polarities for Jupiter and Saturn. The real problem is that many morethan four degrees are needed to well describe even the axisymmetric part of the radial sur-face field vs latitude; the plot of this requires Gauss coefficients up to at least degree 15before the high-latitude peaks become prominent as they are in Fig. 8. For example, thedegree 9 coefficient is −1.087 gauss.

Future missions to Jupiter and Saturn that orbit much closer to their surfaces will providemore accurate estimates of higher degree magnetic structure and might detect a bandedmagnetic field structure as this simulation predicts. Gravity measurements in such missionscould also test the banded cylindrical pattern of differential rotation deep below the surface(Hubbard 1990).

Since the amplitudes of the simulated zonal wind and magnetic field at the surface aresomewhat similar to those of Jupiter and Saturn, it is instructive to check if the amplitudesand 3D geometrical configurations of the flows and fields well below the surface satisfythe constraint in (6) regarding the total rate of entropy production due to ohmic heating.Although time dependent, in general this constraint is satisfied in this simulation because theamplitude of the non-axisymmetric flow decreases with depth and because of the tendencyfor the field to satisfy Ferraro’s iso-rotation law (Ferraro 1937; Glatzmaier 2008).

However, the spatial resolution of this simulation is not sufficient to capture small scaleeddies and vortices near the surface. Actually, no deep convection simulation has yet even


Fig. 9 Snapshots of aBoussinesq simulation of a gasgiant dynamo. Left:Zonally-averaged longitudinalvelocity in the meridian plane.Reds are prograde; blues areretrograde. Right:Zonally-averaged radial magneticfield in the meridian plane. Redsare outward; blues are inward.(Heimpel and Gomez Perez2008)

produced a large vortex at the surface like the “Great Red Spot”. This may require a repre-sentation of radiative transfer and moist convection near the surface, as is done in GCMs.

Another gas giant dynamo model has recently been developed by Heimpel and GomezPerez (2008) based on their Boussinesq deep-convection model. In their model the convec-tion zone represents the outer two-thirds in radius of the planet. Although density stratifi-cation is still neglected, this model has a prescribed electrical conductivity that increasesexponentially by three orders of magnitude from the surface, R, down to radius 0.8 R. Theirelectrical conductivity is constant in the highly-conducting metallic-hydrogen region belowthis radius. An intense magnetic field is maintained in this inner metallic region where theresulting Lorentz forces suppress the zonal wind amplitude (Fig. 9). The dominant featurein the outer low-conductivity region is a strong prograde equatorial jet. The magnetic fieldat the surface of their simulation is dominantly dipolar and much weaker than the field inthe deep interior below 0.8 R.

This simulation nicely demonstrates how an intense magnetic field can inhibit differentialrotation. However, the field in the deep interior may be too intense because the convectiveflows there have amplitudes similar to those near the surface due to the model’s lack ofdensity stratification.

Dynamo models have also been used to investigate the axisymmetry of Saturn’s mag-netic field. Although Jupiter’s (and Earth’s) magnetic fields are dipole dominated, they stillcontain some observable amount of non-axisymmetry. For example the dipole tilts (ratio ofequatorial dipole to axial dipole) for Earth and Jupiter are approximately 10 degrees (Steven-son 1983). In contrast, Saturn observations by Pioneer 11, Voyager I, II and Cassini obtaindipole tilts of < 0.1–1 degrees and a similar lack of non-axisymmetry in the higher mo-ments (Acuna and Ness 1980; Ness et al. 1981; Acuna et al. 1981, 1983; Smith et al. 1980;Davis and Smith 1990; Giampieri and Dougherty 2004). This almost perfect axisymmetryseems to contradict Cowling’s theorem (Cowling 1933) which states that a dynamo cannotmaintain a purely axisymmetric field. However, Cowling’s theorem applies to the dynamosource region, so there is no contradiction if some mechanism can be found that axisym-metrizes the observed field outside the planet.

Stevenson (1980, 1982, 1983) suggested that a helium rain-out layer in Saturn couldproduce such a mechanism. At conditions in Saturn’s interior where hydrogen attains asignificant conductivity, helium becomes immiscible in hydrogen and would therefore rain-out of the hydrogen. This would result in a thin stably-stratified layer surrounding the deepdynamo region. Stevenson proposed that differential rotation in this layer due to thermalwinds could shear out the non-axisymmetry in the field. In Stevenson’s model, the thermalwinds result because of pole to equator temperature differences at Saturn’s surface due to


solar insolation. Kinematic dynamo models have investigated the axisymmetrizing effectsof stably-stratified layers surrounding the dynamo (Love 2000; Schubert et al. 2004). Theyfound that flows in surrounding stable layers do not necessarily act to axisymmetrize thefield. Depending on the interior field morphology and on the geometry of the stable layerzonal flows, a variety of field symmetries were produced.

Dynamic dynamo models have also investigated the effect of surrounding stable lay-ers in an effort to explain Saturn. Christensen and Wicht (2008) found that with a thickstable layer, a highly axisymmetric field can result for certain parameter values. The av-erage dipole tilt in their model is 1.5 degrees. However, their models work with a muchthicker stable layer (∼ 0.4 core radii) than the thin helium-rain out layer initially proposed.Thin stably-stratified layers have also been investigated by Stanley and Mohammadi (2008).These models demonstrated that coupling between the stable and unstable layers can act todestabilize the dynamo and result in more non-axisymmetry. However, these models do notdrive flows in the stable layer via thermal winds the way Stevenson envisioned.

A recent model by Stanley (2008) employs an outer thermal boundary condition on thedynamo model to mimic the effect of laterally varying solar insolation, or alternatively, thethermal perturbations naturally produced by convection in Saturn’s non-metallic outer lay-ers (Aurnou et al. 2008). They find that the surface magnetic fields can be axisymmetrizedthrough these zonal flows, but only when the boundary thermal perturbations result in ther-mal winds matching the sign and morphology of those already occurring in the deeper in-terior. The necessary profile to produce axisymmetrization is the expected profile for solarinsolation or atmospheric convection scenarios. This suggests that Stevenson’s mechanismmay explain the axisymmetrization of Saturn’s magnetic field. Of course, since these mod-els employ the Boussinesq approximation and do not model the outer layers of Saturn, theyare intended more to study the specific mechanism for Saturn axisymmetrization rather thanproduce a complete model of Saturn’s interior flows. In addition, vigorous convection mayeasily overshoot through a thin stably-stratified layer, both from below and from above,completely dominating any thermal wind driving by solar insolation at the surface.

An important question regarding the axisymmetrization of Saturn’s magnetic field andregarding the dynamo mechanism in general for giant planets is how deep do the observedzonal winds extend below the surface, i.e., what is the differential rotation in radius and lati-tude. Liu et al. (2008) attempt to answer this question by doing a steady-state, axisymmetric,kinematic scale-analysis. By estimating the amount of electric current that would be gener-ated within the interior with their prescribed differential rotation shearing their prescribedinternal magnetic field and using their estimate of the depth-dependent electrical conduc-tivity they get an estimate of the total amount of ohmic heating as a function of the depthof the differential rotation. Assuming that this ohmic heating cannot exceed the observedluninosity, they predict that the zonal winds extend no deeper than 0.96 of Jupiter’s radiusand 0.85 of Saturn’s radius.

This is a bold attempt to answer a difficult question, especially without the aid of a 3Dself-consistent convective dynamo simulation. However, their analysis has several uncer-tainties (Glatzmaier 2008; Jones and Kuzanyan 2009). As shown by Backus (1975), Hewittet al. (1975), the total ohmic heating can exceed the luminosity because the bulk of the dissi-pation occurs at a much higher temperature than the surface temperature. Also, as shown byFerraro (1937) for the conditions assumed by Liu et al. (2008), if the internal poloidal fieldis everywhere parallel to surfaces of constant angular velocity there would be no electriccurrent generated and therefore no ohmic heating produced. Of course, some current needsto be generated to maintain a global magnetic field and so the dynamo mechanism is inti-mately connected with the differential rotation below the surface. How well the field obeys


the Ferraro iso-rotation law and allows deep-seated zonal winds in gas giants is still an openquestion.

3.4.2 Ice Giants

The magnetic fields of Uranus and Neptune are non-axially, non-dipolar dominated (see pa-pers by Dougherty and Russell and by Fortney in this issue). It turns out that generatingaxial-dipolar dominated dynamo models is relatively easy when working in an Earth-likethick shell geometry with standard boundary conditions and driving forces. Even observa-tions suggest that axial-dipolar dominated fields are the norm since both Earth and Jupiterhave similar magnetic field morphologies although vastly different interior structure andcomposition. Explaining the ice giants’ anomalous fields has therefore resided in under-standing what could be different inside these planets to result in such strange magnetic fields.

Kinematic dynamo models investigating field symmetries show that axial dipolar, ax-ial quadrupolar, and equatorial dipolar magnetic fields can result from very similar flows,suggesting that it may not be difficult to generate non-axial-dipole dominated fields (Gub-bins et al. 2000). Indeed, although the majority of Earth-like geometry dynamo modelsproduce axially dipolar dominated fields, non-dipolar, non-axisymmetric dynamo modelshave been found (Grote et al. 1999, 2000; Grote and Busse 2000; Ishihara and Kida 2000;Aubert and Wicht 2004). These solutions are found in isolated regions of parameter spaceand it remains unknown whether these solutions would exist in the planetary parameterregime (although one has to admit that this is also the case for the axial-dipole dominateddynamo models). However, there are two fundamental issues in using these models to ex-plain Uranus’ and Neptune’s fields: (i) These non-axial-dipolar dominated dynamos producemuch simpler fields than observed in Uranus and Neptune. They are dominated by a spe-cific component of the large scale field whereas Uranus and Neptune have significant powerin several modes (e.g. dipole, quadrupole, octupole, axisymmetric and non-axisymmetric).(ii) These models do not match the structure of the ice giant interiors determined fromtheir low heat flows (Podolak et al. 1991; Hubbard et al. 1995). It is therefore unclear as towhether these simpler models can explain Uranus’ and Neptune’s fields. However, they arecertainly important in understanding mechanisms of non-dipolar non-axial field generation.

Kutzner and Christensen (2002) found that non-dipolarity increases with Rayleigh num-ber, suggesting that one could explain Uranus and Neptune by saying they convect morevigorously than the other planets. However, based on observations of Uranus’ and Nep-tune’s low heat flows, it seems unlikely that the ice giants are more supercritical than the gasgiants.

Uranus’ and Neptune’s low heat flows resulted in an early proposition to explain theirmagnetic fields. Hubbard et al. (1995) demonstrated that the observed heat flux is incon-sistent with whole-planet convection. They showed that the heat flows imply that only theouter 0.4 Uranus radii and 0.6 Neptune radii of the planets are convectively unstable. Theysuggest that the stably-stratified interior regions can explain the low heat flows and possiblyalso the anomalous magnetic fields since this stratification results in a thin-shell dynamogeometry. The stratification is due to compositional gradients in the interiors of the planetswhich are not unreasonable considering the composition and thermal evolution of the icegiants.

Stanley and Bloxham (2004, 2006) investigated whether dynamos operating in thin layerssurrounding stably-stratified interiors could produce Uranus and Neptune like fields. Theyfound that non-dipolar, non-axisymmetric fields resulted with similar surface power spectrato the observations from these planets. This is due to the influence of the stably stratified inte-rior on the stability of the dipole component in these models. Thin shell models surrounding


insulating solid interiors, conducting solid interiors and stably-stratified fluid interiors wereinvestigated. It was found that, although the fluid flows in these models are relatively similarin morphology and intensity, insulating solid inner core models and stably-stratified inner re-gion models produced magnetic fields with a very different morphology than the conductingsolid inner core models. In the conducting solid inner core models, the anchoring effect dueto the inner core on magnetic field lines stabilized the dynamo producing a strong dipolarfield. When the electrical conductivity of the inner core was removed, this anchoring effectwas reduced, and in combination with the thinner shell geometry, resulted in smaller scale(non-dipolar, non-axisymmetric) fields that reversed continuously. Models with thin shellssurrounding stably-stratified interiors produced similar fields to the insulating solid innercore models because the fluid inner core can mimic an insulating solid core in its responseto electromagnetic stress. The stably-stratified interior loses its anchoring influence becausethe magnetic fields can differentially move the fluid, whereas they cannot do so in a solidconducting interior.

These dynamo models only considered the deep regions of the planet where the pres-sures are high enough such that the electrical conductivity is substantial enough to drivea dynamo. The outer layers of the planet are not modeled. Models by Gomez-Perez andHeimpel (2007) instead consider the outer regions of the planets as well by incorporatinga radially variable electrical conductivity. They find that the deeper regions of the planetswhich have the larger conductivity have much weaker zonal flows than the outer, relativelymore insulating layers. The higher magnetic diffusivities in these models, combined withthe surrounding zonal flows, can result in non-dipolar, non-axisymmetric fields. Similar tothe Stanley and Bloxham (2004) models, these models produce complex fields with similarpower spectra as those of Uranus and Neptune observations. However, they do not adhere tothe interior geometry constrained by the heat flow observations.

3.5 Extrasolar Planet Models

More than 300 extrasolar giant planets have been detected during the past decade us-ing Doppler measurements of the periodic orbital velocities of their sun-like parent stars.A small number have also been detected using photometry during their transits in front ofand behind their parent stars. Extrasolar terrestrial planets, because of their much smallermass, have been much more difficult to detect.

The extrasolar giants that have been discovered have small orbits because of an observa-tional selection effect due to the correspondingly short orbital periods. Having orbital radiias small as 0.05 AU, compared to Jupiter’s 5 AU orbital radius, makes the incident stellarradiation four orders of magnitude greater than that received by Jupiter from our sun. Thesmall orbital radii also mean large tidal effects; so it is assumed that the rotation and orbitalperiods of these planets are nearly synchronous (e.g., Goldreich and Soter 1966). That is,one hemisphere is always facing the parent star and the other is always hidden from it. Thisexternal heating, especially on the dayside, likely produces a very stable thermal stratifica-tion down to much greater pressures in these “Hot Jupiters” than the roughly 1 bar pressurelevel that marks the top of the convection zone in our solar system giants. This extreme day-side heating and nightside cooling must also drive strong atmospheric circulations. Whatstructure these winds have and how deep they extend have been key questions on whichseveral modeling groups have attempted to shed light.

Many modeling studies of the non-magnetic surface dynamics on Hot Jupiters have comefrom the atmospheric climate modeling community. Like the GCMs for solar system giants,these models completely neglect convection in the deep interior and the zonal winds it main-tains. For a nice review of these see Showman et al. (2007).


These models predict wind speeds up to 105 m/s and atmospheric temperatures up to104 K, which means that in some regions these winds can be supersonic. GCMs do notsimulate supersonic flows; nor do anelastic models. This problem requires a model thatsolves the fully-compressible radiative-hydrodynamic equations. The price one pays is thatthe numerical time step needs to be much smaller in order to resolve sound waves. Dobbs-Dixon and Lin (2008) have used such a model to study 3D winds in a Hot Jupiter. Thismodel simulates roughly the outer 8% in radius of the planet and does not make the shallow-water simplifications employed in GCMs. Their model treats the radiation transfer using aflux-limited method that smoothly varies from the diffusion limit (for an optically thickregion) to the streaming limit (for an optically thin region). The simulated winds reach aMach number of 2.7. The zonal wind pattern has a broad prograde equatorial jet and mid-latitude retrograde jets. They argue how important it is for this problem to solve the full3D momentum equation and employ a self-consistent treatment of the radiation transfer.However, they do not address the dynamics of the deep interior.

Evonuk and Glatzmaier (2009) have developed an anelastic model to study convectionand zonal winds of the deep interior of an extrasolar gas giant, assuming a negligible mag-netic field. The model is a fully-convective density-stratified rotating fluid sphere; but themodel’s top boundary is placed below the shallow weather layer. Vigorous convection isdriven throughout the model planet by a heat source in the central region meant to repre-sent the heat generated by the slow contraction of the planet. When the planetary rotation isrelatively weak compared to buoyancy, the preferred mode of convection is a dipolar flowthrough the center of the planet; a greater rotational influence results in strong zonal flowsthat extend deep into the interior. The winds at the model’s surface have a broad progradeequatorial jet and mid-latitude retrograde jets.

What have not yet been studied are the types of magnetic fields that could be generatedin extrasolar gas giants. The proximity of Hot Jupiters to their parent stars suggests thatthe outer atmospheres of these planets are partially ionized and therefore electrically con-ducting. The type of dynamos that operate in these supersonic surface flows is a fascinatingunstudied problem. Ideally, for Hot Jupiters, the dynamos of both the parent star and thegas giant should be simulated simultaneously with robust treatments of the gravitational,radiation and magnetic interactions.

3.6 Future Model Improvements

Future convective dynamo models of giant planets should include more realistic radial strat-ifications of density and electrical conductivity. They should solve the full momentum equa-tion, as done in deep convection models, and should include radiative transfer and moistconvection using schemes near the surface that are at least as self-consistent and sophis-ticated as those employed in GCMs of the Earth’s atmosphere. Higher spatial resolutionwill also be needed so viscosity and thermal conductivity can be reduced and more stronglyturbulent convection can be simulated.

The velocity boundary conditions prescribed in these models also need to be improved.The top boundary should not continue to be a fixed impermeable boundary, which artificiallyforces radial flow into horizontal flow. The vertical coordinate could instead be the columnmass, as in one dimensional planetary evolutionary models, or some variation of pressure,as in GCMs. Alternatively, if one wishes to continue using radius as the vertical coordinate,permeable “free surface” boundaries should be employed within a stratosphere.

The treatment of the very deep interior should also be improved. Some extrasolar giantplanets (and possibly also Jupiter) likely have no solid core or even a stably-stratified fluid


core at some time in their evolution. Models of these planets require a full sphere instead ofa spherical shell. As discussed above, preliminary full-sphere rotating, stratified, convectioncalculations have been produced (Evonuk and Glatzmaier 2009); however, these need to berun as a dynamo to see how the field and flow interact in the deep interior constrained by therate of entropy production by ohmic heating.

Other complications that need to be addressed in simulation studies of giant planets arethe hydrogen phase transition and helium settling (Stevenson and Salpeter 1977; Guillotet al. 2004), double diffusive convection (e.g., Stellmach et al. 2009) and tidal effects andinertial modes (e.g., Guillot et al. 2004; Tilgner 2007).

4 Satellite Dynamos

At least two moons in our solar system have evidence for dynamo action sometime duringtheir history. Observations from the Galileo mission demonstrated that Ganymede has anintrinsic field most likely due to an active dynamo (Kivelson et al. 1996). Data from variousmissions showed that Earth’s Moon, on the other hand, possesses localized crustal magneticfields which may be evidence of a past dynamo (Fuller and Cisowski 1987; Halekas et al.2001). Further information on lunar magnetism comes from the magnetization of Apollosamples allowing constraints to be placed on the timing of the lunar dynamo (Cisowski etal. 1983; Garrick-Bethel et al. 2009). These two bodies maintain some of the most intriguingquestions in planetary magnetic fields.

For Ganymede, the Galileo mission measured a dipole moment, but no information on thehigher multipoles is available. The data can be modeled relatively well by an appropriatelyscaled Earth-dynamo model. However, the question remains as to how a small body likeGanymede can maintain a present-day fluid outer core with strong convection (althoughperhaps this is not as surprising in light of the fact that Mercury does as well). The presenceof sulfur in Ganymede’s core is employed to explain this, and the possibility that the drivingforce for Ganymede is freezing of iron snow below the CMB (Hauck et al. 2006) suggestthat there may be work for dynamo modelers yet. In addition, the orbital resonances of thefour Galilean satellites may provide alternative driving mechanisms for Ganymede in itspast, as well as an additional heat source maintaining Ganymede’s fluid outer core. A mean-field model for Ganymede investigated the effects of Jupiter’s magnetic field in aiding thegeneration of Ganymede’s field by providing a source field for the convective motions tointeract with Sarson et al. (1997). They found that although Jupiter’s seed magnetic field isnot required to maintain Ganymede’s dynamo, it may influence the polarity of the resultingfield.

If the Moon’s remanent magnetic field is the result of rocks cooling in the presence of anancient dynamo (rather than due to impact magnetization, Hood and Artemieva 2008), thenit must contain a metallic core; a piece of information current interior models cannot verify.Thermal evolution models for the Moon suggest that it can maintain a short-lived dynamoearly in its history, or slightly later in its history if the core is insulated early on by a radioac-tive thermal blanket (Stegman et al. 2003). Impact magnetization and demagnetization havelikely played a role is shaping the morphology of the crustal field, but not enough informa-tion is available to constrain the dynamo generated field from the crustal remanent fields,and hence no dynamo models. However, one aspect of the Moon which dynamo modelersshould investigate is the importance of mechanical stirring due to the stronger tidal forces inthe past.


5 Conclusions

Since magnetic fields are generated in a planet’s deep interior and extend beyond the sur-face where they are observable, magnetic fields can act as important probes of planetaryinterior structure and dynamics. Planetary dynamo models therefore, not only investigatethe magnetic field generation process, but also tell us about regions of the planet difficultto study with other means. Current planetary dynamo models have opened a window intoplanetary cores, providing insights into core fluid flows, convective stability and thermalevolution. However, much work is still needed before the planetary dynamo process is fullyunderstood and we can take full advantage of the implications of planetary magnetic fieldobservations.

The main improvements needed are in numerical modeling methods and computationalpower and better observational and experimental data on magnetic fields, interior structureand dynamics. Current and near-future planetary missions such as MESSENGER, Bepi-Colombo, Juno and Grail will hopefully provide us with new constraints on the planetarydynamo process.

Acknowledgement G.G. thanks NASA for support under grants NNG05GG69G and NNX09AD89G. S.S.thanks NSERC for support under the Discovery Grants program.

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